Question: Let $a,$ $b,$ $c,$ $d$ be distinct real numbers such that the roots of $x^2 - 10ax - 11b = 0$ are $c$ and $d,$ and the roots of $x^2 - 10cx - 11d = 0$ are $a$ and $b.$  Find the value of $a + b + c + d.$
Explanation: By Vieta's formulas,
\begin{align*}
c + d &= 10a, \\
cd &= -11b, \\
a + b &= 10c, \\
ab &= -11d.
\end{align*}From the first equation,
\[d = 10a - c.\]From the third equation,
\[b = 10c - a.\]Substituting into the second and fourth equations, we get
\begin{align*}
c(10a - c) &= -11(10c - a), \\
a(10c - a) &= -11(10a - c).
\end{align*}Expanding, we get
\begin{align*}
10ac - c^2 &= -110c + 11a, \\
10ac - a^2 &= -110a + 11c.
\end{align*}Subtracting these equations, we get
\[a^2 - c^2 = 121a - 121c,\]so $(a + c)(a - c) = 121(a - c).$  Since $a$ and $c$ are distinct, we can divide both sides by $a - c,$ to get
\[a + c = 121.\]Hence, $a + b + c + d = 10c + 10a = 10(a + c) = \boxed{1210}.$